The Fourier Transform (FT) is one of the most widely used algorithms in modem science and engineering. In fields such as radar, sonar, radio astronomy, X-ray crystallography, magnetic resonance imaging, tomography, and ultrasound, images can be formed directly by calculating the 2D Fourier Transform of collected signal data to provide a picture based on the reflections of waves transmitted at different frequencies.
The Fourier transform, and also its FFT (Fast Fourier Transform) and DFT (Discrete Fourier Transform) versions, can be used for a variety of purposes, in addition to formation of images. It can be used to reduce noise, to extract or enhance specific features of data, to analyze emission/absorption spectra of materials and distant stars, to describe scattering behavior in radar images, to provide compressed representations of data, to study radar returns (signatures) of stealth aircraft, process medical data (MRI and ultrasound), and to produce 3D terrain images. While the Fourier Transform is used to produce images, unless additional processing occurs (weighting, scaling, clipping, etc.) such images can be noisy and/or contain reverberation-like artifacts called “sidelobes.”
The Fourier series makes it possible to transform the information in one domain (input data) to a different, but equivalent, domain (output image). The Fourier transform frequency, or output domain, contains the same information as the original function, which is in the time domain. The major problem in using the transform is computational, since it requires N2 computations. The Fourier Transform only became widely used after the advent of modern computers and digital computing techniques. A key breakthrough came in 1965, when James Cooley and John Tukey published their algorithm that is now known as the Fast Fourier Transform (FFT). This algorithm requires only N logN calculations, compared to the 2N2 required by the Discrete Fourier Transform (DFT).
The Fast Fourier Transform makes use of symmetries present in the discrete transform calculation; it builds up the final results in stages, each twice as large as the previous one. The creation of the FFT led to an explosion of the FT in engineering applications. Now, the FFT is one of most widely used mathematical calculations in all of engineering and physics, and is used as a common computing benchmark for engineering applications. The DFT is used to form images for synthetic aperture radar, magnetic resonance imaging, sonar, ultrasound, tomography, X-ray crystallography, and different types of spectroscopy, among others.